Let \( f(x)=2 x^{2}+5 x-3, g(x)=x^{2}-4 x+2 \), and \( h(x)=-x^{2}+4 x-2 \). Select all of the expressions that are equivalent to \( 3 x^{2}+x-1 \).
(3 points)
\( f(x)-h(x) \)
\( g(x)-h(x) \)
\( g(x)+f(x) \)
\( h(x)-f(x) \)
\( f(x)+g(x) \)
\( f(x)+h(x) \)
To find which expressions are equivalent to \(3x^2 + x - 1\), we need to set \(3x^2 + x - 1\) equal to each expression and see if they are true:
1) \(f(x) - h(x)\)
Setting \(f(x) - h(x) = 3x^2 + x - 1\), we get:
\((2x^2 + 5x - 3) - (-x^2 + 4x - 2) = 3x^2 + x - 1\)
Simplifying both sides of the equation, we have:
\(2x^2 + 5x - 3 + x^2 - 4x + 2 = 3x^2 + x - 1\)
\(3x^2 + x - 1 = 3x^2 + x - 1\)
This is true, so \(f(x) - h(x)\) is equivalent to \(3x^2 + x - 1\).
2) \(g(x) - h(x)\)
Setting \(g(x) - h(x) = 3x^2 + x - 1\), we get:
\((x^2 - 4x + 2) - (-x^2 + 4x - 2) = 3x^2 + x - 1\)
Simplifying both sides of the equation, we have:
\(x^2 - 4x + 2 + x^2 - 4x + 2 = 3x^2 + x - 1\)
\(2x^2 - 8x + 4 = 3x^2 + x - 1\)
This is not true, so \(g(x) - h(x)\) is not equivalent to \(3x^2 + x - 1\).
3) \(g(x) + f(x)\)
Setting \(g(x) + f(x) = 3x^2 + x - 1\), we get:
\((x^2 - 4x + 2) + (2x^2 + 5x - 3) = 3x^2 + x - 1\)
Simplifying both sides of the equation, we have:
\(x^2 - 4x + 2 + 2x^2 + 5x - 3 = 3x^2 + x - 1\)
\(3x^2 + x - 1 = 3x^2 + x - 1\)
This is true, so \(g(x) + f(x)\) is equivalent to \(3x^2 + x - 1\).
4) \(h(x) - f(x)\)
Setting \(h(x) - f(x) = 3x^2 + x - 1\), we get:
\((-x^2 + 4x - 2) - (2x^2 + 5x - 3) = 3x^2 + x - 1\)
Simplifying both sides of the equation, we have:
\(-x^2 + 4x - 2 - 2x^2 - 5x + 3 = 3x^2 + x - 1\)
\(-3x^2 - x + 1 = 3x^2 + x - 1\)
This is not true, so \(h(x) - f(x)\) is not equivalent to \(3x^2 + x - 1\).
5) \(f(x) + g(x)\)
Setting \(f(x) + g(x) = 3x^2 + x - 1\), we get:
\((2x^2 + 5x - 3) + (x^2 - 4x + 2) = 3x^2 + x - 1\)
Simplifying both sides of the equation, we have:
\(2x^2 + 5x - 3 + x^2 - 4x + 2 = 3x^2 + x - 1\)
\(3x^2 + x - 1 = 3x^2 + x - 1\)
This is true, so \(f(x) + g(x)\) is equivalent to \(3x^2 + x - 1\).
6) \(f(x) + h(x)\)
Setting \(f(x) + h(x) = 3x^2 + x - 1\), we get:
\((2x^2 + 5x - 3) + (-x^2 + 4x - 2) = 3x^2 + x - 1\)
Simplifying both sides of the equation, we have:
\(2x^2 + 5x - 3 - x^2 + 4x - 2 = 3x^2 + x - 1\)
\(x^2 + 9x - 5 = 3x^2 + x - 1\)
This is not true, so \(f(x) + h(x)\) is not equivalent to \(3x^2 + x - 1\).
Therefore, the expressions that are equivalent to \(3x^2 + x - 1\) are:
- \(f(x) - h(x)\)
- \(g(x) + f(x)\)
Answer: \(\boxed{\text{Only }} f(x) - h(x) \text{ and } g(x) + f(x) \text{ are equivalent to } 3x^2 + x - 1.}\)